Simplify; express your answer in exponential form. Assume $y\neq 0, n\neq 0$. $\dfrac{{(y^{-1}n^{3})^{-1}}}{{(y^{-2}n^{-5})^{4}}}$
Solution: To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(y^{-1}n^{3})^{-1} = (y^{-1})^{-1}(n^{3})^{-1}}$ On the left, we have ${y^{-1}}$ to the exponent ${-1}$ . Now ${-1 \times -1 = 1}$ , so ${(y^{-1})^{-1} = y}$ Apply the ideas above to simplify the equation. $\dfrac{{(y^{-1}n^{3})^{-1}}}{{(y^{-2}n^{-5})^{4}}} = \dfrac{{yn^{-3}}}{{y^{-8}n^{-20}}}$ Break up the equation by variable and simplify. $\dfrac{{yn^{-3}}}{{y^{-8}n^{-20}}} = \dfrac{{y}}{{y^{-8}}} \cdot \dfrac{{n^{-3}}}{{n^{-20}}} = y^{{1} - {(-8)}} \cdot n^{{-3} - {(-20)}} = y^{9}n^{17}$